∫ |π^2 xsin(πx)|dx x ∈ to (-1, 3/2). (1) 4π + 1 (2) 3π+1 (3) 5π+1 (4) 6π+1

Question

\(\int_\limits{-1}^{3 / 2}\left|\pi^{2} x \sin (\pi x)\right| d x.\)

(1) \(4 \pi+1\)

(2) \(3 \pi+1\)

(3) \(5 \pi+1\)

(4) \(6 \pi+1\)

Answer 1

Correct option is: (2) \(3 \pi+1\) 

\(I=\int_\limits{-1}^{3 / 2}\left|\pi^{2} x \sin (\pi x)\right| d x\)

\( =\int_\limits{-1}^{1}\left|\pi^{2} x \sin (\pi x)\right| d x+\int_\limits{1}^{3 / 2}\left|\pi^{2} x \sin (\pi x)\right| d x \)

\(=2 \int_\limits{0}^{1}\left|\pi^{2} x \sin (\pi x)\right| d x-\pi^{2} \int_\limits{1}^{3 / 2}|x \sin (\pi x)| d x\)   

\(=2 \pi+1+\pi\)

\(=3 \pi+1\)